Is the pit in Dark Knight Rises a representation of Ra's Al Ghul's Lazarus pit?

At the risk of merely stealing someone else's hypothesis (due in some part to there being no official word on the matter), I would point you to this thread at superherohype.com where some excellent points have been made, including:

the chant which the inmates recite is in Moroccan; can't be coincidental that it is Ra's al Ghul's native language

Talia is the one who is born in it and escapes it, not Bane

The League seems to be the only thing from the outside world that knows the Pit's location

when Bruce was first in the Pit, he was there as a paraplegic but when he left he was fully "healed," albeit not supernaturally

One of the commentators then posted the following image which seems to correlate with the events in the film, but actually depicts a lazarus pit:

I can't take the credit for these ideas (which seem quite conclusive to me), but I am happy to make them available to you ;)

While the pit itself is never given the name of "Lazarus", many other blogs and wikipedia believe it to be a simple homage to this entity in DC's universe.

Further, using the points you have given:

Batman is seen getting over his injury in a short time

• Assuming that the one inmate resets his spine in a few days after Bane has left, he has about 4-5 months after this event in order to fully recover. Realism comes into play here (and as @Nobby has stated), the conditions in which he is in and the time frame makes one wonder if the pit itself has any special healing properties to it. Assuming Nolan's 'realistic' Batman universe where there is nothing supernatural, one can assume this is a simple plot hole.

• Subscribing to the notion that due to this being the finale of Batman, one can also argue that this is Nolan's hint into the supernatural world Batman faces on a daily basis in his comic counterparts.

In the movie he [Bane] is shown as a really strong guy with a mask to subdue his pain from his previously sustained injury.

• This directly counters the above argument. If the Pit itself has properties to help Bruce Wayne overcome a severe back injury and get into enough shape off of very little provisions and effort to best Bane, then why does Bane need the mask in the first place? It can be assumed that he was sitting in the Pit for a long time while Talia summons Ra's to save Bane. It would not be out of the realm of possibility, assuming Bruce's condition that he should be just fine.

• However, we are never told what kind of injury Bane faced and why it was severe enough for a constant flow of anesthetic gas. Also, it can be argued that the Pit only heals those that are righteous in nature, therefore healing Bruce due to his fight against the unjust and criminals, and not Bane who is fighting for a more sinister goal in general (also shot down due to his reason for getting the injury in the first place, protecting the innocence of a child).

All in all, the answer may be as simple as, this is a Pit. It is the only realistic way to introduce Lazarus Pits into Nolan's Batman, which aims for realism of the characters above all else (minus a few sci-fi gadgetry)

As said, without official word from Nolan, we may never know. But looking to Inception, Nolan may never want to give a concrete answer.

I think you are all right in your analysis. But you are missing one key point, the Pit, in the movie heals you so quickly because it brings out your will. Apparently, in the form of fear. It is not uncommon to hear of stories regarding miraculous recovery when a patient has displayed immense, undying will to recover. I think, since Nolan likes his realism in the Batman trilogy, that he was going for this. I firmly believe the Pit is the Nolan equivalent of the Lazarus Pit. To reinforce my idea, Bane is not souped up on Tytan, he was broken and built in the Pit. This gives him the strength we see Batman show in the fight on the courthouse stairs.

I thought this same exact thing while watching the film.

My take on it is that it is the Lazarus Pit. The only problem being that Nolans universe is a realistic one. (ie: Bane does not grow to superhuman size when fed the venom from his suit, as this would be impossible in real life and not very realistic, so they give him a mask which feeds him some sort of pain killer thus making him pretty much invincible.)

In real life, people do not come back from the dead.

So, in comes the underground prison or pit. Life in this pit is pretty much hopeless and one may as well be dead if imprisoned there.

BUT if one could climb out of the pit and escape, as Bruce Wayne did in the film, he could be said to have "Risen from the dead".

I thought the Lazarus Pit analogy in the film was very well done, particularly when the story of Bane unfolds at the end. When Ra's Al Ghul talks of many forms of immortality this definitely re-enforces this idea.